Hausdorff Center for Mathematics Courses

Random Polytopes

Explores beta and beta' polytopes in stochastic geometry, discussing random cones, Poisson cells, and spherical objects. Examines angles and functionals of these models.

• YouTube
• 44 minutes
• Self-Paced
• Free Video

Random Polytopes II

Explore beta and beta' polytopes in stochastic geometry, including random cones, Poisson cells, and spherical objects. Learn to compute expected angles and express model functionals.

• YouTube
• 42 minutes
• Self-Paced
• Free Video

The Large Deviations Approach to High-Dimensional Convex Bodies - Lecture I

Explores large deviation principles in high-dimensional convex bodies, focusing on non-universal behaviors in random projections and their relation to underlying geometry.

• YouTube
• 51 minutes
• Self-Paced
• Free Video

Zakhar Kabluchko - Random Polytopes

Introduction to beta polytopes: convex hulls of samples from beta density on unit ball and beta' density on Euclidean space. Explores models in stochastic geometry reducible to these polytopes.

• YouTube
• 51 minutes
• Self-Paced
• Free Video

A Remark on the Minimal Dispersion

Explore improved upper bounds for minimal dispersion in unit cubes, with some results sharp to logarithmic factors. Gain insights into this mathematical concept.

• YouTube
• 59 minutes
• Self-Paced
• Free Video

Random Tessellations in Hyperbolic Space - First Steps

Explore random tessellations in hyperbolic space, focusing on Poisson hyperplane tessellations and their unique dimension-dependent phenomena compared to Euclidean and spherical spaces.

• YouTube
• 1 hour
• Self-Paced
• Free Video

An Overview on a Young Research Topic - Valuations on Spaces of Functions

Explore valuations on function spaces, from convex bodies to Lipschitz and convex functions. Learn about recent classifications and advancements in this emerging mathematical field.

• YouTube
• 51 minutes
• Self-Paced
• Free Video

The Mysterious Extremals of the Alexandrov-Fenchel Inequality

Explore the Alexandrov-Fenchel inequality's extremal bodies, recent progress in characterizing them for convex polytopes, and insights into this fundamental problem in convex geometry.

• YouTube
• 1 hour 2 minutes
• Self-Paced
• Free Video

Functional Inequalities and Concentration of Measure III

Explores concentration inequalities in probability and geometric analysis, focusing on functional inequalities like Poincaré and log-Sobolev. Covers properties, applications to Lipschitz functions, and various measure settings.

• YouTube
• 49 minutes
• Self-Paced
• Free Video

Gaussian Isoperimetry and Related Topics III

Explore Gaussian isoperimetry's applications in probability and analysis, including concentration inequalities, noise stability, and Sobolev-type inequalities. Learn proof methods from geometric measure theory.

• YouTube
• 49 minutes
• Self-Paced
• Free Video

Some Applications of Variational Techniques in Stochastic Geometry III

Explores probabilistic bounds on Poisson space, measuring Gaussian distance for two-scale stabilization elements. Discusses quantitative CLTs for weakly stabilizing functionals and novel bounds for strongly stabilizing ones.

• YouTube
• 50 minutes
• Self-Paced
• Free Video

Some Applications of Variational Techniques in Stochastic Geometry II

Explore second-order Poincaré inequalities on the Poisson space and their geometric applications, focusing on quantitative CLTs and fourth-moment theorems in stochastic geometry.

• YouTube
• 48 minutes
• Self-Paced
• Free Video

Functional Inequalities and Concentration of Measure II

Explores concentration inequalities in probability and geometric analysis, focusing on functional inequalities like Poincaré and log-Sobolev. Covers properties, applications to Lipschitz functions, and various measure settings.

• YouTube
• 48 minutes
• Self-Paced
• Free Video

Gaussian Isoperimetry and Related Topics II

Explore Gaussian isoperimetry's applications in probability and analysis, including concentration inequalities, noise stability, and Sobolev-type inequalities. Learn proof methods from geometric measure theory.

• YouTube
• 47 minutes
• Self-Paced
• Free Video

On Yuansi Chen's Work on the KLS Conjecture II

Explores recent progress on the KLS conjecture in high-dimensional geometry, focusing on Yuansi Chen's work and Eldan's Stochastic Localization technique. Discusses implications for Bourgain's slicing problem.

• YouTube
• 46 minutes
• Self-Paced
• Free Video